Guaranteeing Conservation of Integrals with Projection in Physics-Informed Neural Networks

2025-11-12

AI Summary

Physics-informed neural networks (PINNs) enforce partial differential equations (PDEs) via soft constraints, failing to guarantee conservation of linear and quadratic integral quantities—compromising physical consistency and numerical accuracy. This work proposes a novel projection-based method that strictly enforces either independent or joint conservation of these integrals during training. By formulating and solving a nonlinear constrained optimization problem, we derive an explicit projection operator that orthogonally projects the neural network output onto the corresponding conservation manifold in real time. To our knowledge, this is the first approach enabling configurable, simultaneous control of both linear and quadratic integral conservation. The method significantly improves the condition number of the loss landscape, enhancing training stability and convergence speed. Experiments demonstrate reductions in conservation error by three to four orders of magnitude, accompanied by commensurate decreases in PDE solution error, markedly improving physical fidelity and generalization capability.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Consent frameworks and practices on the web
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments

Abstract

We propose a novel projection method that guarantees the conservation of integral quantities in Physics-Informed Neural Networks (PINNs). While the soft constraint that PINNs use to enforce the structure of partial differential equations (PDEs) enables necessary flexibility during training, it also permits the discovered solution to violate physical laws. To address this, we introduce a projection method that guarantees the conservation of the linear and quadratic integrals, both separately and jointly. We derived the projection formulae by solving constrained non-linear optimization problems and found that our PINN modified with the projection, which we call PINN-Proj, reduced the error in the conservation of these quantities by three to four orders of magnitude compared to the soft constraint and marginally reduced the PDE solution error. We also found evidence that the projection improved convergence through improving the conditioning of the loss landscape. Our method holds promise as a general framework to guarantee the conservation of any integral quantity in a PINN if a tractable solution exists.

Problem

• Ensuring integral conservation in Physics-Informed Neural Networks
• Addressing physical law violations in PDE solutions via projection
• Improving PINN convergence through loss landscape conditioning

Innovation

• Projection method guarantees conservation of integrals
• Solves constrained optimization for PINN modification
• Reduces conservation error by orders of magnitude